Valentin Féray ; Ekaterina A. Vassilieva
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Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result
dmtcs:2815 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
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https://doi.org/10.46298/dmtcs.2815
Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result
Authors: Valentin Féray 1; Ekaterina A. Vassilieva 2
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Valentin Féray;Ekaterina A. Vassilieva
1 Laboratoire Bordelais de Recherche en Informatique
2 Laboratoire d'informatique de l'École polytechnique [Palaiseau]
We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.
A bijective proof of Jackson's formula for the number of factorizations of a cycle
1 Document citing this article
Source : OpenCitations
Bernardi, Olivier; Chapuy, Guillaume, 2011, A Bijection For Covered Maps, Or A Shortcut Between Harer–Zagierʼs And Jacksonʼs Formulas, Journal Of Combinatorial Theory, Series A, 118, 6, pp. 1718-1748, 10.1016/j.jcta.2011.02.006.